Stir together parsley, oil, lemon juice, garlic and thyme. Stir in salt and pepper. Set pesto aside while grilling mushrooms (or chill pesto up to 2 hr.).
Brush all sides of mushroom caps with some of the oil floating on the pesto. Place caps, gill-side down, on BBQ preheated to medium-high. Grill 1 to 2 min. or until marked. Flip over. Grill 1 to 2 min., or until mushrooms are tender and juices start to pool in caps.
Carefully transfer (avoid tipping out juices) mushrooms to serving platter. Spoon pesto into caps. Serve mushrooms warm.
Alternatively, cook mushrooms in skillet on stovetop on medium-high heat.
Use store-bought pesto.